Atomic branching in molecules

Atomic Branching in Molecules

ERNESTO ESTRADA,1 JUAN A. RODRIGUEZ-VELAZQUEZ,2

MILAN RANDIC1

1Complex Systems Research Group, X-Rays Unit, RIAIDT, Edificio CACTUS, University ofSantiago de Compostela, 15782 Santiago de Compostela, Spain2Department of Mathematics, University Carlos III de Madrid, 28911 Leganes, Madrid, Spain

Received 10 June 2005; accepted 1 August 2005Published online 1 November 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20850

ABSTRACT: A graph theoretic measure of extended atomic branching is definedthat accounts for the effects of all atoms in the molecule, giving higher weight to thenearest neighbors. It is based on the counting of all substructures in which an atomtakes part in a molecule. We prove a theorem that permits the exact calculation of thismeasure based on the eigenvalues and eigenvectors of the adjacency matrix of thegraph representing a molecule. The definition of this measure within the context of theHuckel molecular orbital (HMO) and its calculation for benzenoid hydrocarbons arealso studied. We show that the extended atomic branching can be defined using anyreal symmetric matrix, as well as any Hermitian (self-adjoint) matrix, which permits itscalculation in topological, geometrical, and quantum chemical contexts. © 2005 WileyPeriodicals, Inc. Int J Quantum Chem 106: 823–832, 2006

Key words: atomic complexity; extended valence; spectral graph theory; self-returning walks; spectral moments

Introduction

M any properties of an atom in a moleculeare determined mainly by its nearest-

neighbor atoms [e.g., nuclear magnetic resonance(NMR) chemical shifts, atomic charges]. This isone of the reasons we refer in chemistry, forinstance, to carbon atoms as primary, secondary,tertiary, or quaternary, depending on the numberof substituents that are different from hydrogenof the corresponding atom. A quaternary carbonatom is, for instance, a carbon atom bonded tofour other atoms different from hydrogen; i.e., it

has four nearest neighbors. This terminology re-fers to the degree of branching of an atom in amolecule, understanding it as the number of at-oms that are different from hydrogen bonded toan atom; incidentally, this is known in graphtheory as the degree of a vertex or graph theoret-ical valence [1]. Many properties depend on theatomic branching of a carbon atom. Thus, forexample, Wiener [2] was prompted to search for aquantitative structure–property relationship byqualitative statements found in chemical texts ofthe day that the boiling points of alkanes arerelated to branching of their molecular skeletons.No attempts were presented in those early daysto describe or characterize molecular branching.Similarly, it is well known that the stability of theCorrespondence to: E. Estrada; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 106, 823–832 (2006)© 2005 Wiley Periodicals, Inc.

carbocations generated from primary, secondary,tertiary, and quaternary carbon atoms are of adifferent order of magnitude [3]. However, theproperties of atoms in a molecule are also af-fected by the second nearest-neighbor atoms, i.e.,the atoms bonded to its nearest neighbors. Take,for instance, a primary carbon atom that isbonded to a quaternary, a tertiary or a secondarycarbon. In the first case, it has only one secondnearest neighbor, but this number increases up tofour in the last case affecting its atomic propertiesin a different way. This situation demonstratesthat the concept of atomic branching based onlyon the nearest neighbors of an atom is not appro-priate to describe a number of atomic properties.This has been the motivation of the use of aug-mented valence as a complexity index for mole-cules [4 – 6].

This idea of augmented vertex (atom) branch-ing extends beyond atoms in molecules and isalso applicable to other systems represented bycomplex networks [7–10]. Although in a complexnetwork most effects are transmitted from oneelement to another through the link joining themtogether, it is obvious that this effect can reachtopologically distant elements by following apath of inter-element links [11, 12]. Thus, if weconsider two elements with the same number ofconnections, the one having the larger number ofsecond nearest neighbors will have a greaterchance of receiving an information flow throughthe network than the other. The “value” of thisinformation can decreases as the length of thepath increases (think of “value” in terms of timedelay), which make closest neighbors more im-portant than the more distant ones. This is wellknown by organic chemists who have studied thetransmission of inductive effects through molec-ular backbones to explain molecular reactivity[13].

In the search for such extended atomic branch-ing we desire a property that accounts for theeffects of all atoms in the molecule giving higherweight to the closest ones. It is also desired thatthis property can be generalized to different rep-resentations of a molecule and not be limited toone or another of such representations, e.g.,graph-theoretic, quantum chemical, geometric. Inthe present work, we introduce a measure ofextended atomic branching that fulfill these re-quirements. It is based on counting all substruc-tures in which an atom takes part in a molecule.We prove a theorem that permits the exact calcu-

lation of this measure based on the eigenvaluesand eigenvectors of the adjacency matrix of thegraph representing this molecule. The definitionof this measure within the context of the Huckelmolecular orbital (HMO) and its calculation forbenzenoid hydrocarbons is also studied. Weshow that the extended atomic branching can bedefined using any real symmetric matrix, andshow examples of this generalization to geomet-ric matrices. In the last section of this work weprove that the extended atomic branching can bedefined by using any Hermitian (self-adjoint) ma-trix, which permits its calculation in quantumchemical contexts.

Extending Atomic Branching toLong-Range Contributions

In the intuitive and simple idea of branching ofan atom i, defined as the number of atoms differentof hydrogen bonded to i, we only consider thenumber �0 of its closest neighbors: B(i) � �0. Inother words, �0 is the number of bonds involvingthe corresponding atom i; i.e., its vertex degree �i ingraph theoretic terminology. If we want to gener-alize this approach by considering all subgraphs inwhich the vertex i participates, we have to use amore general expression for atomic branching, suchas

EB�i� � �0 � �1�1 � �2�2 � · · · � �n�n, (1)

where the terms �k�k can be considered as correc-tion factors accounting for longer-range effects tothe branching of the vertex i. The average over allthe vertices’ branching can be used as a criterion ofmolecular branching. This idea follows the pioneer-ing work of Ruch and Gutman [14], who intro-duced partial ordering of graphs according to the“degree of branching,” which was then extended byMichalski [15] discussing possible connections be-tween branching extent and spectra of trees. An-other version based on partial order was also ana-lyzed by Klein and Babic [16], which is in some wayrelated to our current “higher-order” extension ofthe concept of branching.

Mathematically, vertex degrees can be obtainedas the main diagonal elements of the second powerof the adjacency matrix of the graph [17]. Conse-quently, the original definition of branching of anatom can be expressed as follows:

ESTRADA, RODRIGUEZ-VELAZQUEZ, AND RANDIC

824 VOL. 106, NO. 4

(2)

In general, the ith diagonal entry of the kthpower of the adjacency matrix can be expressed asa linear combination of different subgraphs con-taining the vertex i [18]. For instance, the ith ele-ment of the main diagonal of the third power of the

adjacency matrix counts twice the number of trian-gles containing vertex i and the ith diagonal ele-ment of the fourth power of the adjacency matrixcounts the number of times vertex i takes place incertain subgraphs as follows:

(3)

. (4)

Then, if we take a linear combination of thesediagonal entries of the adjacency matrix and ar-range them properly, we can express branchingdefined by expression (1) in terms of powers of theadjacency matrix:

EB�i� � C0�A0�ii � C1�A� � C2�A2�ii � C3�A3�ii

� C4�A4�ii � · · · � Ck�Ak�ii. (5)

Substituting the expressions of main diagonalentries by their linear combinations in terms ofsubgraphs, we have

(6)

where the term in brackets accompanying the coef-ficient C0 is equal to one, i.e., the atom i, and thecoefficient C1 is only present in the case where the

graph has weighted vertices. After appropriate re-arrangement we obtain expression (1), now writtenin graphic form:

(7)

ATOMIC BRANCHING IN MOLECULES

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 825

where the coefficients �i are combinations of coef-ficients used in (6).

We can think about this process as an optimizationin which we transform an atomic branching indexbased only on contributions coming from adjacentatoms into another that also considers contributions

from “higher-order” substructures. For instance, if weconsider the graph representing 1,2,3-trimethylpen-tane, which has five vertices of degree one and threevertices of degree three, and calculate the first 10powers of the adjacency matrix we obtain the vertexdegrees shown below at the right:

Here we used the expression like (5) with k � 10.We can select the coefficients such that larger sub-structures receive less weight in the linear combi-nation than the smaller ones. For mathematical con-venience (see further discussion), we have selectedthese coefficients as Ck � 1/k!, obtaining the aboveresults for 1,2,3-trimethylpentane.

Extended Atomic Branching:Mathematical Formulation

Mathematically, the terms (Ak)ii represent thenumber of closed walks (CWs) of length k, alsoknown as self-returning walks, that start and end atthe atom i [19, 20]. A walk of length k is a sequenceof (not necessarily different) atoms �1, �2, . . . , �r,�r�1, such that for each i � 1, 2, . . . , N there is a linkfrom �i to �i�1. A closed walk (CW) is a walk inwhich �r�1 � �1 [20]. The number of CWs starting(and ending) at atom i can be expressed in terms ofthe spectral properties of the molecular graph [21]:

�k�i� � �Ak�ii � �j�1

N

��j�i��2��j�k, (8)

where �j is the jth eigenvalue of the adjacency ma-trix and �j(i) is the ith component of the eigenvectorassociated with the jth eigenvalue.

It is then easy to realize that the approach wepreviously used to extend the atomic branching inorder to account for larger substructures in themolecule is given by

EB�i� � �k�0

��k�i�

k! . (9)

Let �1 be the main (the principal or the leading)eigenvalue of A. For any non-negative integer k andany i � {1, . . . , N }, �k(i) �1

k, series ¥k�0� [�k(i)/k!],

whose terms are non-negative, converges:

EB�i� �k�0

��1

k

k! � e�1. (10)

The following result shows that EB(i) can beobtained mathematically from the spectra of theadjacency matrix of the molecular graph [22].

Theorem 1. Let G � (V, E) be a simple moleculargraph of order N. Let �1, �2, . . . , �N be an orthonor-mal basis of �N composed of eigenvectors of Aassociated with the eigenvalues �1, �2, . . . , �N. Let�j(i) denote the ith component of �j. For all i � V, theatomic extended branching may be expressed asfollows:

EB�i� � �j�1

N

��j�i��2e�j. (11)

Proof. The orthogonal projection of the unit vec-tor ei (the ith vector of the canonical base of �N) on�j is

pj�ei� ��ei, �j

��j�2 �j � �ei, �j�j � �j�i� � �j. (12)

Hence, the number of CWs starting at atom i canbe expressed in terms of the spectral properties ofthe graph as follows:


826 VOL. 106, NO. 4

�k�i� � �Ak�ii � �Akei, ei � �Ak �j�1

N

pj(ei), �j�1

N

pj(ei)��

j�1

N

��j�k��j�i��2. (13)

Then, using our expression defining the ex-tended atomic branching EB(i) � ¥k�0

� [�k(i)/k!], weobtain

EB�i� � �k�0

� ��j�1

N (�j)k[�j(i)]2

k! � . (14)

By reordering the terms of series (11), we obtainthe absolutely convergent series:

�j�1

N � [�j(i)]2 �k�0

� (�j)k

k! � � �j�1

N

��j�i��2e�j, (15)

which obviously also converges to EB(i). Thus, theresult follows and expression (11) is an exact for-mula for calculating the extended branching of anatom represented by a molecular graph.

Extended Atomic Branching in theHMO Context

We begin by considering conjugated moleculesusing the HMO approach [23]. The normalized lin-ear combination of atomic orbitals (LCAO) wavefunction of the jth molecular orbital (MO) of the-electronic system of a conjugated hydrocarbonwith N carbon atoms is given by

�j � �j�1

N

Cj�i��i. (16)

The energy of the jth orbital is the expectationvalue for the one-electron effective Hamiltonian H:

j � ��j�H��j. (17)

By assuming equal values, respectively, for allthe Coulomb, and all the nonzero resonance inte-grals, the energy of ith orbital is expressed as

j � � � ��j, (18)

where �j is the jth eigenvalue of the adjacency ma-trix, A, of the molecular graph representing thehydrogen-depleted skeleton of the hydrocarbon.The total -electron energy satisfies the relation

E � �j�1

N

��j�. (19)

The kth spectral moment of the adjacency matrixof the molecular graph is then

�k � �j�1

N

��j�k � �

�

��

xk�� x�dx, (20)

where �(x) � ¥j�1N �(x �j) is the spectral density

functional, and �(x) is the Dirac function. A physi-cal interpretation for a CW in a conjugated hydro-carbon can be provided by an analogy with thespectral moments of the spectral density in a Hub-bard model in a lattice [24]. A CW of length k ischaracterized by the following steps: (i) remove anelectron from atom 0� leaving a hole, (ii) move thehole by exchanging its place with adjacent electronsthat carry their spins with them; and (iii) after ksteps, replace the first electron (with its spin) in thehole at atom R� � 0�, completing the closed walk. Thefinal spin configuration must then coincide with theinitial one. Thus, the infinite sum ¥k�0

� �k(i) corre-sponds to the number of times in which a -elec-tron at atom i moves back and forward visiting theother atoms in the molecule. We will consider thatthe larger this number, the larger the electron den-sity at the corresponding atom as a consequence ofthe “longer permanence” of this electron at thecorresponding atom. It is also straightforward torealize that the “topological time” in which theelectron completes a CW is directly related to itslength. Thus, the shorter the CW, the shorter thetime in which the electron returns to the atom,which immediately implies a longer permanence ofthe electron in such atom. This intuition indicatesthat the shorter CWs will have more influence inthe electron density of atom n than the larger ones,which correspond exactly to our formulation of theextended valence using the expression

�k�0

��k�i�

k! .



In the HMO context with � � 0 and � � 1, theextended atomic branching is given by

EB�i� � �j�1

N

�Cj�i��2e j. (21)

This formula can also be expressed in terms ofoccupied and vacant molecular orbitals:

EB�i� � �j

OCC

�Cj�i��2e j � �j

VAC

�Cj�i��2e j. (22)

Extended Atomic Branching forBenzenoid Hydrocarbons

A CW is called odd (even) if k is odd (even). It isalso easy to show that

EB�i� � �j�1

N

�Cj�i��2cosh� j� � �j�1

N

�Cj�i��2sinh� j�

� EBeven�i� � EBodd�i�, (23)

which means that the term EBodd(i) only accountsfor substructures containing at least one odd cycle.Benzenoid polycyclic aromatic hydrocarbons donot contain odd cycles, and consequently their mo-lecular graphs are bipartite [25]. A graph is bipartiteif its vertex set V can be partitioned into two subsetsV1 and V2, such that all edges have one endpoint inV1 and the other in V2. In such cases, it is obviousthat

EB�i� � EBeven�i� � �j�1

N

�Cj�i��2cosh� j�. (24)

In the case of bipartite molecules, such as benze-noid hydrocarbons, it is well known that the spec-trum is symmetric, i.e., j � Nj�1 [25]. We willthen have the following expression for the extendedatomic branching:

EB�i� � 2 �j

OCC

�Cj�i��2cosh� j�. (25)

Table I gives the values of the extended valencefor the carbon atoms of seven benzenoid hydrocar-

bons (see Fig. 1). We have previously consideredthat a CW in a conjugated hydrocarbon can repre-sent the moving back and forward of the -elec-

TABLE I ______________________________________Values of extended atomic branching calculated bythe graph-theoretic method in the present work andby extended Huckel method and DFT calculations.

Compound EB(i) EHT Ab initio

Benzene 2.2800 2.032Naphthalene

1 2.2846 2.0432 2.3387 2.0523 3.1409 2.943

Phenanthrene1 3.2004 2.9602 2.3405 2.0573 2.2846 2.0414 2.2846 2.0445 2.3387 2.0496 3.1428 2.9547 2.3405 2.054

Anthracene1 2.2846 2.042 1.9612 2.3387 2.048 1.9743 3.1428 2.947 2.9524 2.3947 2.074 2.014

Pyrene1 3.1447 2.9522 2.3406 2.0453 2.3389 2.0544 2.2863 2.0345 3.2623 2.978

Chrysene1 2.3405 2.0592 3.2004 2.9623 3.2023 2.9684 3.2023 2.9685 2.3423 2.0646 2.3406 2.0577 3.1428 2.9538 2.3387 2.0509 2.2846 2.04410 2.2846 2.042

Naphthacene1 2.2846 2.0422 2.3387 2.0503 3.1428 2.9524 2.3947 2.0785 3.1447 2.951

R 0.998 0.998


828 VOL. 106, NO. 4

trons through the carbon–carbon skeleton. In thiscase, it is justified to express the extended branch-ing as the sum of two terms, one coming from thenearest neighbors of atom i and the other comingfrom long-range contributions. We can then con-sider the first term as the number of conjugatedbonds incident to atom i, i.e., atom degree, and thesecond term as the correction introduced by theelectron motion along the different longer substruc-tures of the molecule. By full analogy with thedefinition of atomic charge density, we will writeEB(i) � B(i) q(i), where B(i) � �i is the number ofconjugated bonds incident to i, and q(i) is a type oftopological charge produced by the motion of-electrons through the carbon–carbon skeleton ofthe molecule.

As a consequence of this interpretation, wehave calculated the extended Huckel theory(EHT) extended valences for the same carbon

atoms as [26]: EBEHT(i) � B(i) qEHT(i). For thecase of anthracene, we have also defined an abinitio extended valence using DFT calculations atcc-pVDZ B3LYP level reported by Dessent [27]:EbDFT(i) � B(i) qDFT(i). As can be seen in bothcases the extended atomic branching definedfrom graph spectra shows very good linear cor-relations with those obtained from quantumchemical atomic charges, which indicates that, atleast for polycyclic benzenoid hydrocarbons, theextended branching brings important electronicand topological molecular information.

Extended Atomic Branching BeyondSimple Graphs

In the previous section, we have limited ourdiscussion to a purely graph theoretic contextbased on the use of the adjacency matrix of amolecular graph. However, it would be of greattheoretical and practical importance to expandthe concept of extended atomic branching beyondthe use of adjacency matrix and to account forother graph theoretic as well as quantum chemi-cal matrices. Theorem 1 is also true for real sym-metric matrices (the proof is similar to the onegiven below and will be not shown here). Thus,we will illustrate a couple of examples of extend-ing the concept used to develop characterizationof atomic branching for this kind of matrices. The

TABLE II ______________________________________Values of the extended branching index calculatedfrom the weighted adjacency matrices of n-butane,2-butene, and 2-butyne.

Weighted adjacency matrix Extended atomic branching

A � 0 1.54 0 0

1.54 0 1.54 00 1.54 0 1.540 0 1.54 0

A �

0 1.52 0 01.52 0 1.34 0

0 1.34 0 1.520 0 1.52 0

A �

0 1.46 0 01.46 0 1.20 0

0 1.20 0 1.460 0 1.46 0

FIGURE 1. Molecular structures of polycyclic aro-matic benzenoid compounds studied.



first corresponds to a weighted graph in whichedges are weighted by bond lengths. Let rij be thebond distance for the bond between atoms i andj. The elements of the adjacency matrix of theweighted graph are then defined as follows:

aij � �rij if i and j are adjacent0 otherwise.

Table II illustrates the adjacency matrices and theextended atomic branching for the carbon atoms ofn-butane, 2-butene, and 2-butyne. It can be seenthat the values decrease from the alkane to thealkyne for both the methyl group and the carbonwith different hybridizations: Csp3

Csp2 Csp. It is

worth noting that this approach permits the differ-entiation of heteroatoms in the molecular structureand it is also applicable in the HMO context forhetero-conjugated molecules [23].

The second example given here is concerned withthe so-called D/D matrices [28]. These matrices aredefined for molecular graphs of given fixed geometry.Matrix element (i, j) is given as the quotient of theEuclidean (through space) distance between atoms iand atom j and the graph theoretical distance betweenthe same two atoms (through bonds), which is givenby the number of bonds between the atoms. Theleading eigenvalue of the D/D matrix in the case ofchain structures has been interpreted as a measure ofthe degree of chain folding, because in more foldedstructures there will be a larger number of smallermatrix elements, resulting in a smaller magnitude ofthe leading eigenvalue. Besides being used for char-acterization of small molecules, D/D matrices and thedegree of folding of proteins [29, 30] D/D matrices

offered a way for the characterization of general non-chain structures [31], including polyhedral structures[32] and even more realistic molecular models basedon representation of atoms by overlapping sphereshaving van der Waals radii [33]. More recently, D/Dmatrices have been used not only for numerical char-acterizations of DNA in which individual DNA se-quences are represented by a set of matrix invariantsbut also for numerical characterization of proteomicsmaps [34].

Table III illustrates the D/D matrices and the ex-tended atomic branching for two conformational iso-mers of 1,3-butadiene, namely s-cis and s-trans iso-mers. The values of the extended branching for thecarbon atoms are lower for the s-cis isomer that for thes-trans, which also coincides with the previous exam-ple, in which the molecules with shorter bond dis-tances showed smaller EB(i) values. These values dif-fer significantly from the values of EB(i) obtainedbefore from the adjacency matrices. The main idea ofthis section is that we can extend the definition ofextended atomic branching to any representation ofthe graph by means of real symmetric matrices.

Extended Atomic Branching FromHermitian Matrices

A square matrix is called Hermitian if it is self-adjoint. Therefore, a Hermitian matrix A � (aij) isdefined as one for which A � AH, where AH de-notes the conjugate transpose. Hereafter, �a� and a�indicate the modulus and the complex conjugate ofa. Thus, in a Hermitian matrix, aij � a�ji and theirdiagonal elements are real. It is of common use to

TABLE III _____________________________________________________________________________________________Values of the extended branching index calculated from the D/D matrices for two conformational isomers of1,3-butadiene.

s-cis isomer s-trans isomer

D/D � 0 1.342 1.179 0.933

1.342 0 1.344 1.1791.179 1.344 0 1.3420.933 1.179 1.342 0

D/D � 0 1.342 1.179 1.198

1.342 0 1.344 1.1791.179 1.344 0 1.3421.198 1.179 1.342 0


830 VOL. 106, NO. 4

represent observables by Hermitian matrices/oper-ators in quantum mechanics. We recall that Hermi-tian matrices are unitarily diagonalizable, and theyhave real eigenvalues. Then, the use of Hermitianoperators/matrices to represent observables guar-antees that the outcome of observations is real [35].

For a graph G � (V, E), we define a complexfunction � : V � V 3 C such that, f(u, u) � R, f(u,�) � 0 if and only if u and � are adjacent, and f(u,�) � f(�, u). In such conditions we define the Her-mitian adjacency matrix of G as Hij � f(�i, �j). So, thematrix H is Hermitian, justifying the terminologyused.

That is, the Hermitian extended branching ofatom i in the molecule is given by

EB�i� � �k�0

� �Hk�ii

k! . (26)

Let � be the main eigenvalue of H. For anynon-negative integer k and any i � {1, . . . , n},�k(i) �k, series (2), whose terms are non-negative,converges:

�k�0

� �Hk�ii

k! �k�0

��k

k! � e�. (27)

Thus, the Hermitian extended valence of any atomi is bounded above by EB(i) e�.

Theorem 2. Let H be the Hermitian (adjacency)matrix associated with the graph G � (V, E) oforder N. Let �1, �2, . . . , �N be an orthonormal basisof CN composed of eigenvectors of H associatedwith the eigenvalues �1, �2, . . . , �N. Let �j

i denotethe ith component of �j. For all i � V, the Hermitianextended branching can be expressed as follows:

EB�i� � �j�1

N

��ji�2e�j. (28)

Proof. The orthogonal projection of the unit vec-tor ei (the ith vector of the canonical base of Rn) on�j is

prj�ei� ��ei, �j

��j�2 �j � �ei, �j�j � �ji � �j. (29)

Hence,

�Hk�ii � �Hkei, ei � �Hk �j�1

N

prj(ei), �j�1

N

prj(ei)��

j�1

N

�jk��j

i�2. (30)

Using expression (26), we obtain

EB�i� � �k�0

� ��j�1

N�j

k��ji�2

k! � . (31)

By reordering the terms of series (31), we obtainthe absolutely convergent series:

�j�1

N � �ji 2 �

k�0

��j

k

k!� � �j�1

N

��ji�2e�j, (32)

which obviously also converges to EB(i).In the present work, we will not provide any

calculation result for the application of these matri-ces in defining the extended atomic branching in acontext of quantum chemistry, as these calculationswill be detailed in a separate publication. However,it is straightforward to realize the new avenues thatTheorem 2 opens for extending this context in aquantum chemical context.

Conclusions

We have introduced a measure that extend theconcept of atomic branching beyond the traditionalconsideration of the nearest-neighbor atoms. Theidea is that the atomic branching of an atom de-pends not only on its closest neighbors, but also onatoms that are separated at topological distanceslarger than one. The influence of an atom on theatomic branching of another decreases as the num-ber of bonds separating them increases. The way inwhich we introduce this atomic branching permitsits exact calculation from the spectrum of the adja-cency matrix of the molecular graph. However, wehave shown that this approach is applicable toother molecular representations, which include theuse of Hermitian matrices as those used in quantumchemistry to represent observables.

The atomic branching index developed hereshow some interesting and desirable characteristics.It is known that in complex systems the relation-



ships between elements are both short range andlong range. One atom receives “information” fromnear neighbors by means of short-range direct localinteractions. However, this information can travelfrom one atom to another, having an indirect influ-ence on all other atoms. This characteristics is wellaccounted for by the atomic branching index, whichtakes into account both short-range and long-rangeinteractions in a hierarchical way, in which the firstreceive more weight than the second. The othercharacteristic of the branching degree that meritssome attention is that it is not obtained as thesimple sum of the different parts of the system, butit is defined on the basis of global topological prop-erties of the molecule. In a complex system, there isa sense that the different elements cannot “know”what is happening in the system as a whole. On thecontrary, all the complexity of the system would bededuced from the complexity of such single ele-ment. In other words, the molecular complexity iscreated by the interrelationships between the atomsand not by a simple sum of individual atomic prop-erties. Both characteristics suggest that the aug-mented atomic branching is a good characterizationof the level of complexity of an atom, a character-istic that will be analyzed in future work.

References

1. Trinajstic, N. Chemical Graph Theory; CRC Press: Boca Ra-ton, FL, 1983.

2. Wiener, H. J. Am Chem Soc 1947, 69, 17.3. Olah, G. A.; Surya Prakash, G. K., Eds. Carbocation Chem-

istry; John Wiley & Sons: New York, 2004.4. Randic, M. Croat Chem Acta 2001, 74, 683.5. Randic, M.; Plavsic, D. Croat Chem Acta 2002, 75, 107.6. Randic, M.; Plavsic, D. Int J Quantum Chem 2003, 91, 20.7. Estrada, E. Adv Quantum Chem 2005 (in press).

8. Barabasi, A. L. Linked: The New Science of Networks;Plume: New York, 2003.

9. Buchanan, M. Nexus, W. W. Norton: New York, 2002.10. Bornholdt, S.; Schuster, H. G., Eds. Handbook of Graphs and

Networks: From the Genome to the Internet; Wiley-VCH:Berlin, 2003.

11. Newman, M. E. J. SIAM Rev 2003, 45, 167.12. Holme, P. Phys Rev E 2005, 71, 046119.13. Exner, O. J Phys Org Chem 1999, 12, 265.14. Ruch, E.; Gutman, I. J Comb Inform System Sci 1979, 4, 285.15. Michalski, M. Publ Inst Math (Beograd) 1986, 39, 35.16. Klein, D. J.; Babic, D. J Chem Inform Comput Sci 1997, 37,

656.17. Biggs, N. Algebraic Graph Theory; Cambridge University

Press: Cambridge, UK, 1993.18. Balasubramanian, K. Comput Chem 1985, 9, 43.19. Bonchev, D.; Kier, L. B. Int J Quantum Chem 1993, 46, 635.20. Harary, F. Graph Theory; Addison-Wesley: Reading, 1969.21. Cvetkovic, D.; Rowlinson, P.; Simic, S. Eigenspaces of

Graphs; Cambridge University Press: Cambridge, UK, 1997.22. Estrada, E.; Rodrıguez-Velazquez, J. A. Phys Rev E 2005, 71,

056103.23. Streitweiser, A. Molecular Orbital Theory for Organic Chem-

ists; John Wiley & Sons: New York, 1961.24. Roulet, B.; Fischer, M. E.; Doniach, S. Phys Rev B 1973, 7, 403.25. Gutman, I.; Polanski, O. E. Mathematical Concepts in Or-

ganic Chemistry; Springer-Verlag: Berlin, 1986.26. Hoffmann, R. J Chem Phys 1963, 39, 1397.27. Dessent, C. Chem Phys Lett 2000, 330, 180.28. Randic, M.; De Alba, L. M.; Kleiner, A. F. J Chem Inform

Comput Sci 1994, 34, 277.29. Randic, M.; Krilov, G. Chem Phys Lett 1997, 272, 115.30. Randic, M.; Krilov, G. Int J Quantum Chem 1997, 65, 1065.31. Randic, M. Int J Quantum Chem: Quantum Biol Symp 1995,

22, 61.32. Randic, M.; Krilov, G. Int J Quantum Chem: Quantum Biol

Symp 1996, 23, 1851.33. Randic, M.; Krilov, G. Int J Quantum Chem 1997, 65, 1065.34. Randic, M. In Conn, P. M., Ed.; Handbook of Proteomic

Methods; Humana Press: Totowa, NJ, 2003; p 429.35. Arfken, H. J.; Weber, H. J.; Weber, H.-J. Mathematical Meth-

ods for Physics; Academic Press: San Diego, CA, 1985.


832 VOL. 106, NO. 4

Atomic branching in molecules

Documents

Transcript of Atomic branching in molecules